In this paper, we are concerned with the asymptotic properties and numerical analysis of the solution to hybrid stochastic differential equations with jumps. Applying the theory of M-matrices, we will study the $ p $th moment asymptotic boundedness and stability of the solution. Under the non-linear growth condition, we also show the convergence in probability of the Euler-Maruyama approximate solution to the true solution. Finally, some examples are provided to illustrate our new results.
Citation: |
S. Albeverio
, Z. Brzezniak
and J. Wu
, Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients, J. Math. Anal. Appl., 371 (2010)
, 309-322.
doi: 10.1016/j.jmaa.2010.05.039.![]() ![]() ![]() |
|
W. J. Anderson, Continuous-Time Markov Chains, Springer, Berlin, 1991.
doi: 10.1007/978-1-4612-3038-0.![]() ![]() ![]() |
|
D. Applebaum, Levy Processes and Stochastic Calculus,
Cambridge University Press, 2004.
doi: 10.1017/CBO9780511755323.![]() ![]() ![]() |
|
D. Applebaum
and M. Siakalli
, Asymptotic stability of stochastic differential equations driven by Levy noise, J. Appl. Probab., 46 (2009)
, 1116-1129.
doi: 10.1239/jap/1261670692.![]() ![]() ![]() |
|
J. Bao
, B. Bottcher
, X. Mao
and C. Yuan
, Convergence rate of numerical solutions to SFDEs with jumps, J. Comput. Appl. Math., 236 (2011)
, 119-131.
doi: 10.1016/j.cam.2011.05.043.![]() ![]() ![]() |
|
M. Baran
, Approximations for solutions of Levy-Type Stochastic Differential Equations, Stochastic Analysis and Applications., 27 (2009)
, 924-961.
doi: 10.1080/07362990903136447.![]() ![]() ![]() |
|
N. Bruti-Liberati
and E. Platen
, Strong approximations of stochastic differential equations with jumps, J. Comput. Appl. Math., 205 (2007)
, 982-1001.
doi: 10.1016/j.cam.2006.03.040.![]() ![]() ![]() |
|
A. Gardon
, The order of approximations for solutions of Ito-type stochastic differential equations with jumps, Stoch. Anal. Appl., 22 (2004)
, 679-699.
doi: 10.1081/SAP-120030451.![]() ![]() ![]() |
|
D. J. Higham
and P. Kloeden
, Numerical methods for nonlinear stochastic differential equations with jumps, Numer. Math., 101 (2005)
, 101-119.
doi: 10.1007/s00211-005-0611-8.![]() ![]() ![]() |
|
L. Hu
, X. Mao
and Y. Shen
, Stability and boundedness of nonlinear hybrid stochastic differential delay equations, Syst. Control. Lett., 62 (2013)
, 178-187.
doi: 10.1016/j.sysconle.2012.11.009.![]() ![]() ![]() |
|
L. Hu
, X. Mao
and L. Zhang
, Robust stability and boundedness of nonlinear hybrid stochastic differential delay equations, IEEE Trans. Automa. Control., 58 (2013)
, 2319-2332.
doi: 10.1109/TAC.2013.2256014.![]() ![]() ![]() |
|
J. Jakubowski
and M. Nieweglowski
, Jump-diffusion processes in random environments, J. Differential Equations., 257 (2014)
, 2671-2703.
doi: 10.1016/j.jde.2014.05.052.![]() ![]() ![]() |
|
R. Z. Khasminskii, Stochastic Stability of Differential Equations, Stijhoff and Noordhoff, Alphen, 1980.
![]() ![]() |
|
H. Kunita
, Stochastic diffrential equations based on Lévy processes and stochastic flows of diffomorphisms in Real and Stochastic Analysis, New Perspectives, Berlin, (2004)
, 305-373.
![]() ![]() |
|
X. Li
, X. Mao
and Y. Shen
, Approximate solutions of stochastic differential delay equations with Markovian switching, J. Difference Equ. Appl., 16 (2010)
, 195-207.
doi: 10.1080/10236190802695456.![]() ![]() ![]() |
|
L. Liu
, Y. Shen
and F. Jiang
, The almost sure asymptotic stability and pth moment asymptotic stability of nonlinear stochastic differential systems with polynomial growth, IEEE Trans. Automa. Control., 56 (2011)
, 1985-1990.
doi: 10.1109/TAC.2011.2146970.![]() ![]() ![]() |
|
J. Luo
and K. Liu
, Stability of infinite dimensional stochastic evolution equations with memory and Markovian jumps, Stochastic Process. Appl., 118 (2008)
, 864-895.
doi: 10.1016/j.spa.2007.06.009.![]() ![]() ![]() |
|
X. Mao
, LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 236 (1999)
, 350-369.
doi: 10.1006/jmaa.1999.6435.![]() ![]() ![]() |
|
X. Mao
, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl., 268 (2002)
, 125-142.
doi: 10.1006/jmaa.2001.7803.![]() ![]() ![]() |
|
X. Mao
and M. Rassias
, Khasminskii-type theorems for stochastic differential delay equations, Stoch. Anal. Appl., 23 (2005)
, 1045-1069.
doi: 10.1080/07362990500118637.![]() ![]() ![]() |
|
X. Mao and C. Yuan,
Stochastic Differential Equations with Markovian Switching, Imperial College, London, 2006.
doi: 10.1142/p473.![]() ![]() ![]() |
|
X. Mao, Stochastic Differential Equations and their Applications, Horwood, Chichester, 1997.
![]() ![]() |
|
X. Mao
, Numerical solutions of stochastic differential delay equations under the generalized Khasminskii-type conditions, Appl. Math. Comput., 217 (2011)
, 5512-5524.
doi: 10.1016/j.amc.2010.12.023.![]() ![]() ![]() |
|
G. Marion
, X. Mao
and E. Renshaw
, Convergence of the Euler shceme for a class of stochastic Differential Equations, International Mathematical Journal., 1 (2002)
, 9-22.
![]() ![]() |
|
M. Milosevic
, Existence, uniqueness, almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the Euler-Maruyama approximation, Appl. Math. Comput., 237 (2014)
, 672-685.
doi: 10.1016/j.amc.2014.03.132.![]() ![]() ![]() |
|
E. Mordecki
, A. Szepessy
and R. Tempone
, Adaptive weak approximation of diffusions with jumps, SIAM Journal on Numerical Analysis., 46 (2008)
, 1732-1768.
doi: 10.1137/060669632.![]() ![]() ![]() |
|
B. Oksendal and A. Sulem, Applied Stochastic Control of Jump Diffusions, Springer, Berlin, 2005.
![]() ![]() |
|
E. Platen and N. Bruti-Liberati,
Numerical Solution of Stochastic Differential Equations with Jumps in Finance, Springer, Berlin, 2010.
doi: 10.1007/978-3-642-13694-8.![]() ![]() ![]() |
|
V. Popov, Hyperstability of control system, Springer, Berlin, 1973.
![]() ![]() |
|
S. T. Rong,
Theory of Stochastic Differential Equations with Jumps and Applications, Springer, Berlin, 2005.
![]() ![]() |
|
M. Song
, L. Hu
and X. Mao
, Khasminskii-Type theorems for stochastic functional differential equations, Discrete Contin. Dyn. Syst. Ser. B., 18 (2013)
, 1697-1714.
doi: 10.3934/dcdsb.2013.18.1697.![]() ![]() ![]() |
|
I. S. Wee
, Stability for multidimensional jump-diffusion processes, Stochastic Process. Appl., 80 (1999)
, 193-209.
doi: 10.1016/S0304-4149(98)00078-7.![]() ![]() ![]() |
|
F. Wu
and S. Hu
, Suppression and stabilisation of noise, Internat. J. Control., 82 (2009)
, 2150-2157.
doi: 10.1080/00207170902968108.![]() ![]() ![]() |
|
F. Wu
and S. Hu
, Stochastic suppression and stabilization of delay differential systems, International Journal of Robust and Nonlinear Control., 21 (2011)
, 488-500.
doi: 10.1002/rnc.1606.![]() ![]() ![]() |
|
F. Wu
and S. Hu
, The LaSalle-type theorem for neutral stochastic functional differential equations with infinite delay, Discrete and Continuous Dynamical Systems, 32 (2012)
, 1065-1094.
![]() ![]() |
|
F. Xi
, On the stability of a jump-diffusions with Markovian switching, J. Math. Anal. Appl., 341 (2008)
, 588-600.
doi: 10.1016/j.jmaa.2007.10.018.![]() ![]() ![]() |
|
F. Xi
, Asymptotic properties of jump-diffusion processes with state-dependent switching, Stoch. Process. Appl., 119 (2009)
, 2198-2221.
doi: 10.1016/j.spa.2008.11.001.![]() ![]() ![]() |
|
F. Xi
and G. Yin
, Almost sure stability and instability for switching-jump-diffusion systems with state-dependent switching, J. Math. Anal. Appl., 400 (2013)
, 460-474.
doi: 10.1016/j.jmaa.2012.10.062.![]() ![]() ![]() |
|
Z. Yang
and G. Yin
, Stability of nonlinear regime-switching jump diffusion, Nonlinear Anal., 75 (2012)
, 3854-3873.
doi: 10.1016/j.na.2012.02.007.![]() ![]() ![]() |
|
G. Yin and C. Zhu, Hybrid Switching Diffusion: Properties and Applications,
Springer, New York, 2010.
doi: 10.1007/978-1-4419-1105-6.![]() ![]() ![]() |
|
G. Yin
and F. Xi
, Stablity of regime-switching jump diffusions, SIAM J. Control Optim., 48 (2010)
, 4525-4549.
doi: 10.1137/080738301.![]() ![]() ![]() |
|
S. You
, W. Mao
, X. Mao
and L. Hu
, Analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients, Appl. Math. Comput., 263 (2015)
, 73-83.
doi: 10.1016/j.amc.2015.04.022.![]() ![]() ![]() |
|
C. Yuan
and W. Glover
, Approximate solutions of stochastic differential delay equations with Markovian switching, J. Comput. Appl. Math., 194 (2006)
, 207-226.
doi: 10.1016/j.cam.2005.07.004.![]() ![]() ![]() |
|
C. Yuan
and J. Bao
, On the exponential stability of switching-diffusion processes with jumps, Quart. Appl. Math., 71 (2013)
, 311-329.
doi: 10.1090/S0033-569X-2012-01292-8.![]() ![]() ![]() |
|
S. Zhou
, M. Xue
and F. Wu
, Robustness of hybrid neutral differential systems perturbed by noise, Journal of Systems Science and Complexity, 27 (2014)
, 1138-1157.
doi: 10.1007/s11424-014-2037-9.![]() ![]() ![]() |
|
Q. Zhu
, Asymptotic stability in the pth moment for stochastic differential equations with Levy noise, J. Math. Anal. Appl., 416 (2014)
, 126-142.
doi: 10.1016/j.jmaa.2014.02.016.![]() ![]() ![]() |